Place a medium-size skillet over medium-high heat with about a quarter inch of water. Add the sausages and EVOO to the pan and bring the liquid to a bubble. Cook sausages in boiling water, turning occasionally, until the water has evaporated, about 10 minutes. Once the water has reduced away, let the sausage casings crisp in the pan until golden brown, about 4 minutes.
While the sausages are cooking up, place the rolls onto a baking sheet, cut side up, and toast under the broiler until golden brown. While they're toasting up, place a small pot over medium heat with the butter and crushed garlic. Melt the butter and then brush it over the toasted rolls. Sprinkle the Parmigiano over the cut sides of the roll and place them back under the broiler to melt and lightly brown the cheese.
To serve, split the sausages open lengthwise like a book and lay one into each roll. Top with a scoopful of the slaw salad.